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On the Imaginary Part of the Characteristic Function On the imaginary part of the characteristic function Saulius Norvidas Institute of Data Science and Digital Technologies, Vilnius University, Akademijos str. 4, Vilnius LT-04812, Lithuania (e-mail: [email protected]) Abstract Suppose that f is the characteristic function of a probability measure on the real line R. In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part ℑ f ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that ℑ f ≡ ℑg but f 6≡ g? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts. Keywords: Locally compact Abelian group; Fourier transform; characteristic function; Fourier algebra. Mathematics Subject Classification: Primary 43A35; Secondary 42A82; 60E10. 1 Introduction Let B(R) denote the usual σ-algebra of all Borel subsets of the real line R, and let M(R) be the Banach algebra of bounded regular complex-valued Borel measures µ on R. M(R) is equipped with the usual total variation norm kµk. We define the Fourier (Fourier-Stieltjes) transform of µ ∈ M(R) by µˆ (ξ ) = e−iξxdµ(x), ξ ∈ R. R Z The family of these functions µˆ forms the Fourier-Stieltjes algebra B R . The norm in B R is arXiv:2009.03960v1 [math.CA] 8 Sep 2020 ( ) ( ) defined by kµˆ kB(R) = kµkM(R). If µ ∈ M(R) is nonnegative measure and kµk = 1, then in the language of probability theory, µ and f (ξ ) := µˆ (−ξ ), ξ ∈ R, are called a probability measure and its characteristic function, respectively. In this paper, we deal with several examples and certain assertions demonstrating the relationship between the characteristic function f and its imaginary part ℑ f . It is well known that f is not determined by | f |. More precisely, there exist two different real-valued characteristic functions f and g such that | f | = |g| everywhere (see [4, p. 506] and [11, p. 265]). Next, any characteristic 1 function f 6≡ 1 is never determined by its real part ℜ f , i.e., there exists the characteristic function g such that ℜ f ≡ ℜg but f 6≡ g (see [11, p. 259]). In this context is natural to ask whether the same is true for f and ℑ f . The following question is given in [11, p. 334] as an unsolved problem: Is it true that the characteristic function is never determined by its imaginary part? (1.1) In other words, is it true that for any characteristic function f , there exists a characteristic function g such that ℑ f ≡ ℑg but f 6≡ g? It is not difficult to check that the answer to this question is no. For example, if ℑ f (x) = ℑg(x) = sinx, then necessarily f (x) = g(x) = eix. This can be verified directly by simple arguments. However, we refer to our theorem 1. We note that if f is the characteristic function of a probability measure µ ∈ M(R), then 1 1 ℜ f (x) = eixt + e−ixt dµ(t) and ℑ f (x) = e−ixt − eixt dµ(t). 2 R 2i R Z Z Therefore, both ℜ f and ℑ f are elements of B(R). Moreover, ℜ f is an even characteristic func- tion with kℜ f kB(R) = 1. On the other hand, ℑ f is odd, kℑ f kB(R) ≤ 1, and ℑ f is positive definite if and only if ℑ f ≡ 0. We study the question (1) in the more general case of the characteristic function defined on an arbitrary locally compact abelian group G (see, e.g., [6], [10], and the next section of this paper for the exact definitions of algebras B(G), M(G), B(G), and other background of harmonic analysis on groups). Let us start with a question: which a function ϕ : G → R might serve as the imaginary part of the characteristic function f on G? Keeping in mind what was said above about the imaginary part of the characteristic function, the following theorem gives a complete description of these functions ϕ. Theorem 1.1 Assume that ϕ ∈ B(G) is real-valued and odd. If kϕkB(G) ≤ 1, then there exists a characteristic function f on G such that ℑ f ≡ ϕ. As a consequence, we obtain a characterization of what characteristic functions are uniquely determined by their imaginary parts. Theorem 1.2 Let f : G → C be a characteristic function. Then ℑ f completely determines f if and only if kℑ f kB(G) = 1. The following corollary can be used to construct characteristic functions which are uniquely determined by their imaginary parts. We denote by G the dual group of G (see the next section for the exact definition). For U ⊂ G, we write −U = {t ∈ G : −t ∈ U}. b b 2 Corollary 1.3 Suppose that f : G → C is the characteristic function of a probability measure µ ∈ M(G). Let U ∈ B(G), and assume thatU ∩ (−U) = /0. If b b µ(U) = 1, (1.2) then f is completely determined by ℑ f . Now it is easy to verify that several frequently used characteristic functions on the classical groups R , Z, T, and Rn are uniquely determined by their imaginary parts (see, e.g., a list of several probability distributions and their characteristic functions in [11, p.p. 282-329]). At the end of this chapter, we recall some distributions and their characteristic functions of this type. Example 1.4 The group G = R. (A) The characteristic functions of the following probability distributions are uniquely deter- mined by their imaginary parts: the arcsine distribution, the Bessel distribution, the beta distribution, the gamma distribution, the hyperexponential distribution, the standard Levy distribution, the Maxwell distribution, the Pareto distribution, and the χ2-distribution with density function 1 t(n/2)−1e−t/2, t > 0, p(t) = 2n/2Γ(n/2) ( 0, otherwise, where n is a positive integer. (B) The characteristic functions of the following probability distributions are not uniquely de- termined by their imaginary parts: the normal distribution, the Laplace distribution, and the Cauchy distribution. (C) The characteristic functions f of the uniform distribution and the triangular (Simpson) distribution with density defined by 1 , a ≤ t ≤ b, a < b, p(t) = b−a ( 0, otherwise, and by 4 b−a −|t − a+b | , a ≤ t ≤ b, a < b, p(t) = (a−b)2 2 2 ( 0, otherwise, respectively, are uniquely determined by ℑ f if and only if 0 6∈ [a,b]. Example 1.5 The groups G = Rn,n > 1. The characteristic functions of the multivariate distributions on Rn inherit in most cases the same properties as the appropriate univariates distributions. For example, the characteristic function of the Dirichlet distribution on Rn, i.e., the multivariate generalization of the beta distribution (see, e.g., [7, Chapter 49]), and multivariate Pareto distributions of the first and the second 3 kinds (see, e.g., [1, Chapter 6]) are uniquely determined by their imaginary parts. On the other hand, the characteristic functions of the normal distribution, the Laplace distribution, and the Cauchy distribution are not uniquely determined by their imaginary parts. Example 1.6 The group G = T. In these examples we consider the circle group T as the interval T = R/(2πZ)=[0,2π) with addition mod 2π and with the usual topology inherit from R. Then T = Z. The Bochner theorem [6, p. 293] states now that a function f : T → C is the characteristic function of a probability b measure on Z if and only if ikx f (x) = ∑ αke , (1.3) k∈Z where αk ≥ 0 and ∑k αk = 1. Let A denote the set of all k ∈ Z in (3) such that αk > 0. Then (3) is uniquely determined by its imaginary part if and only if A ∩ (−A) = /0. Therefore, the char- acteristic functions of the negative binomial distribution, the Poisson distribution, the binomial distribution, and the hypergeometric distribution are not uniquely determined by their imaginary parts. Example 1.7 The group G = Z. In this case, the set of characteristic functions on Z coincides with the family of positive definite sequences {ξk : ξk ∈ Z} of complex numbers ξk such that ξ0 = 1. Since Z = T, we see that any such a sequence is the Fourier coefficients sequence of a probability distribution on T. In other b words, this {ξk : ξk ∈ Z} coincides with the Fourier coefficients sequence of a periodic distribu- tion on R with period 2π. A periodic distribution can be obtained by the periodization ( by the ”wrapping” ) around the unit circle T = R/(2πZ) of a distribution on R (see, e.g., [9, Chapter 3]). For example, the characteristic functions f of the wrapped Cauchy distribution, the wrapped normal distribution, and he wrapped exponential distribution are not uniquely determined by ℑ f. On the other hands, according to the corollary 1.3, it is easy to create the probability distribu- tions on T such that their characteristic functions, i.e., the characteristic sequences are uniquely determined by their imaginary parts. 2 PRELIMINARIES We recall and introduce some terminology and notation. Throughout G will be a locally compact abelian group with dual group G consisting of the continuous homomorphisms (or characters) γ : G → T, where T is the multiplicative unit circle group in C.
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